# Chap6 - CHAP 6 FINITE ELEMENTS FOR PLANE SOLIDS FINITE...

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1 CHAP 6 FINITE ELEMENTS FOR PLANE SOLIDS FINITE ELEMENT ANALYSIS AND DESIGN Nam-Ho Kim and Bhavani Sankar Instructor: Subrata Roy

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2 INTRODUCTION • Plane Solids – All engineering problems are 3-D. It is the engineer who approximates the problem using 1-D (beam or truss) or 2- D (plane stress or strain). – Stress and strain are either zero or constant in the direction of the thickness. – System of coupled second-order partial differential equation – Plane stress and plane strain: different constraints imposed in the thickness direction Plane stress : zero stresses in the thickness direction (thin plate with in-plane forces) Plane strain : zero strains in the thickness direction (thick solid with constant thickness, gun barrel) – Main variables: u ( x -displacement) and v ( y -displacement)
3 TYPES OF 2D PROBLEMS • Governing D.E. • Definition of strain • Stress-Strain Relation – Since stress involves first-order derivative of displacements, the governing differential equation is the second-order x y 2 dx xx x σ + 2 dx xx x 2 dy yy y + 2 dy yy y 2 dy yx y τ + 2 dy yx y 2 dx xy x + 2 dx xy x b x b y 0 0 xy xx x xy yy y b xy b τ σ τσ ++ = ∂∂ = ,, xx yy xy uv u v x yy x εεγ ⎛⎞ == = + ⎝⎠ 11 12 13 21 22 23 31 32 33 {} [] {} xx xx yy yy xy xy CC C C C σε τγ ⎧⎫ ⎡⎤ ⎪⎪ ⎢⎥ =⇔ = ⎨⎬ ⎣⎦ ⎩⎭ σ C ε

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4 TYPES OF 2D PROBLEMS cont . • Boundary Conditions – All differential equations must be accompanied by boundary conditions S g is the essential boundary and S T is the natural boundary g : prescribed (specified) displacement (usually zero for linear problem) T : prescribed (specified) surface traction force Objective : to determine the displacement fields u (x, y) and v (x, y) that satisfy the D.E. and the B.C. ,o n n g T S S = = ug nT σ
5 PLANE STRESS PROBLEM • Plane Stress Problem: – Thickness is much smaller than the length and width dimensions – Thin plate or disk with applied in-plane forces – z-directional stresses are zero at the top and bottom surfaces – Thus, it is safe to assume that they are also zero along the thickness – Non-zero stress components: σ xx , σ yy , τ xy – Non-zero strain components: ε xx , ε yy , ε xy , ε zz x y f x f y 0 zz xz yz σττ ===

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6 PLANE STRESS PROBLEM cont . • Stress-strain relation – Even if ε zz is not zero, it is not included in the stress-strain relation because it can be calculated from the following relation: • How to derive plane stress relation? –S o l v e f o r ε zz in terms of xx and yy from the relation of σ zz = 0 and Eq. (1.57) –W r i t e σ xx and σ yy in terms of xx and yy 2 1 2 10 { } [ ] { } 1 00 ( 1 ) xx xx yy yy xy xy E σ σν ε ε σ ε ν τν γ ⎡⎤ ⎧⎫ ⎪⎪ ⎢⎥ =⇔ = ⎨⎬ ⎩⎭ ⎣⎦ C () zz xx yy E ν εσ σ =− +
7 PLANE STRAIN PROBLEM • Plane Strain Problem – Thickness dimension is much larger than other two dimensions.

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Chap6 - CHAP 6 FINITE ELEMENTS FOR PLANE SOLIDS FINITE...

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